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No, you must have copies of original certificate issued after graduation (certificate, diploma or degree certificate).
If you have a Bachelor of Arts or Bachelor of Science degree and you want to apply to be a TSC teacher then you must also have a post graduate diploma in Education. Since you already have a degree, you can enroll to any university that offers post graduate diploma in education like Moi, MKU, KU etc.
If you are referring to a college or university graduation certificate, then you must have one before applying. A secondary school certificate will not help.
As at writing this answer, it is not possible to use Huduma number for TSC number application. This is because the system has not been updated to accept huduma number.
Other than Bachelor of Education, TSC has a list of speific degree courses for Bachelor of Arts and Bachelor of Science graduates that are allowed for one to be registered as a TSC teacher. In addition to being on the specific course list, you are also required to have a post-graduate diploma in education (PGDE) if you did Bachelor of Arts or Bachelor of Science degree.